Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to include transitions occurring at finite energies, beyond the ground state. These transitions can be induced not only by varying a control parameter but also by changing the energy of the system. In this work, we explore ESQPTs in spinor Bose-Einstein condensates, highlighting their experimentally observable signatures. We identify a topological order parameter that distinguishes between excited-state phases, which can be probed through interferometry. Additionally, we discuss a recent experimental realization of these ideas. Our findings open a way for experimental characterization and exploration of excited-state quantum phases in atomic many-body systems.

The cornerstone of modern quantum metrology is the quantum Cramér-Rao bound and the quantum Fisher information. Under generic conditions, this bound can be saturated by an optimal estimator and measurement, provided that a large amount of repeated measurements on the system are performed. However, in the presence of smaller data sets it typically largely underestimates the error that can actually be achieved. In this talk, we present a family of generalized bounds on the variance of unbiased estimators that are larger than the quantum Cramér-Rao bound when the sample is small and thereby provide a more realistic limit on the achievable precision of a finite-sample quantum measurement. In the large-data limit, the hierarchy of bounds collapses back onto the quantum Cramér-Rao bound. 

Quantum metrology develops techniques that improve the measurement resolution of parameters encoded in quantum states. The quantum states of light or of trapped atomic ensembles are defined not only by the quantum state itself, but also by the shape of the modes that this state occupies. Parameters that determine, for example, a quantum optical mode include the spatial and temporal shape as well as the frequency spectrum. The estimation of these parameters is of high interest for applications in imaging, timing, positioning and precision spectroscopy. In this talk, we present a quantum theory for the metrological estimation of mode parameters. We demonstrate that the population of suitably designed modes with nonclassical states enables quantum enhancements for the estimation of arbitrary mode parameters. We discuss applications of our results in the context of superresolution imaging and displacement sensing.

The sensitivity of quantum states under small perturbations is the quantity of central interest in quantum metrology. Besides identifying strategies that lead to quantum-enhanced measurement precision, the metrological sensitivity provides detailed information about the state's quantum correlations. We use a metrological complementarity relation to formulate a witness for Einstein-Podolsky-Rosen (EPR) steering that is stronger than uncertainty-based criteria. We further derive a hierarchy of conditions that bound the multipartite entanglement of local hidden states (LHS) that may account for the observed correlations. Metrological steering criteria can be optimized under experimentally motivated constraints and have particular advantages in detecting non-Gaussian steering.

The well-known squeezing coefficient efficiently quantifies the sensitivity and entanglement of Gaussian states. However, this coefficient is insufficient to characterize the much wider class of non-Gaussian quantum states that can generate even larger sensitivity gains. In this talk, we present a non-Gaussian extension of spin squeezing based on reduced variances of nonlinear observables that can be optimized under relevant constraints. We determine the scaling of the sensitivity enhancement that is made accessible from increasingly complex quantum states generated by one-axis-twisting in the presence of relevant noise processes. Our analytical results provide recipes for optimal non-Gaussian spin squeezing in atomic experiments.

The central tool of quantum metrology, the quantum Fisher information (QFI), quantifies the sensitivity of quantum states under small perturbations. Besides identifying entangled states that can beat classical precision limits, this provides a versatile tool to detect and quantify multipartite entanglement and steering. We show how the QFI gives rise to entanglement witnesses that reveal the multipartite structure of inseparable partitions, generalizing entanglement depth and k-separability. Furthermore, the QFI describes a complementarity relation that can be used to formulate the Einstein-Podolsky-Rosen (EPR) paradox in the framework of quantum metrology, leading to a witness for EPR steering. Metrological entanglement witnesses are more powerful than variance-based methods such as Reid's criterion or spin-squeezing coefficients and can be systematically optimized from a limited set of measurable observables.

Superresolution techniques based on intensity measurements after a spatial mode decomposition can overcome the precision of diffraction-limited direct imaging. However, realistic measurement devices always introduce finite crosstalk in any such mode decomposition. Here, we show that any nonzero crosstalk leads to a breakdown of superresolution when the number N of detected photons is large. Combining statistical and analytical tools, we obtain the scaling of the precision limits for weak, generic crosstalk from a device-independent model as a function of the crosstalk probability and N. The scaling of the smallest distance that can be distinguished from noise changes from N−1/2 for an ideal measurement to N−1/4 in the presence of crosstalk.

The spin squeezing coefficient reveals the multiparticle entanglement depth of an atomic system from collective measurements. The entanglement of addressable modes is typically detected with different methods that require local measurements on the modes. In this talk, we address the question, how much mode entanglement can we expect to generate by distributing a spin-squeezed ensemble into accessible modes. We show that under the assumption of indistinguishability, we can relate the spin-squeezing coefficient to well-known conditions for mode entanglement. Moreover, we point out the relation between different criteria for multiparticle entanglement based on the spin-squeezing coefficient.

We propose a trapped-ion quantum simulator as a platform for the study of electron transfer processes. In particular, we demonstrate how a driven trapped-ion system with controlled dissipation and electron-phonon interactions can be operated in parameter regimes of rich molecular charge transfer physics. Continuous controllability of the system parameters allows us to probe the transition between different transfer mechanisms, including classical and quantum regimes. This allows us not only to test widely-used transport models from chemistry, such as Marcus theory, but also to probe complex and largely unexplored regimes that are often unattainable in molecular experiments.

The well-known squeezing coefficient efficiently quantifies the sensitivity and entanglement of Gaussian states. However, this coefficient is insufficient to characterize the much wider class of highly sensitive non-Gaussian quantum states. In this talk, we present an extension of spin or quadrature squeezing based on reduced variances of nonlinear observables. An analytical optimization of the measurement observable under experimental constraints further allows us to identify those observables that will yield the highest achievable sensitivity and the strongest criterion for entanglement. Our results can be used to identify optimal quantum-enhanced phase estimation protocols and entanglement witnesses for increasingly complex quantum states.

Multiparameter quantum metrology develops strategies to simultaneously estimate several parameters with quantum-enhanced precision and has potential applications in imaging and field sensing. The multiparameter sensitivity is quantified by the covariance matrix of all parameters. We present sensitivity limits for a multimode interferometer as matrix bounds for the covariance matrix. Quantum strategies to improve the precision may consist in entanglement among the parameter-encoding modes or among the particles that enter the interferometer (if their number is fixed). We observe a stepwise enhancement of the achievable precision limit as more modes and particles are entangled. We further discuss the optimal states for the various quantum and classical strategies.

We present upper limits for the sensitivity of separable states which can be applied to continuous-variable systems, leading to an experimentally usable witness for entanglement in a multi-mode displacement sensor. In this talk, we discuss both the theoretical background and the analysis of experimentally generated Gaussian continuous-variable quantum states in photonic systems with homodyne measurements. Moments up to second order provide sufficient information on Gaussian quantum states. Generally, the information available from these moments can be used to define lower bounds on the quantum Fisher information, determined by the covariance matrix. The bounds have a natural interpretation in terms of squeezing, in analogy to spin squeezing coefficients known from discrete variable systems. We show that the lower bounds obtained from continuous-variable squeezing coefficients are saturated by all pure and mixed Gaussian states. Finally, we explore quantum enhancements of the precision for displacement sensing with a single mode. We discuss how a phase-insensitive precision measurement of the displacement amplitude can be realized using Fock states, presenting both theoretical results and an experimental realization with a trapped ion.

We analyze families of measures for the quantum statistical speed which include as special cases the quantum Fisher information, the trace speed, i.e., the quantum statistical speed obtained from the trace distance, and more general quantifiers obtained from the family of Schatten norms. These measures quantify the statistical speed under generic quantum evolutions and are obtained by maximizing classical measures over all possible quantum measurements.

We develop a multi-configurational mean-field method to reproduce semiclassical, spectral features of a family of spin chain models with variable range in a transverse magnetic field. The model includes the Lipkin-Meshkov-Glick model and the Ising model as special cases. The semiclassical spectrum is exact in the limit of very strong or vanishing external magnetic fields. Each of the semiclassical energy landscapes shows a bifurcation when the external magnetic field exceeds a threshold value. This reflects the quantum phase transition from the symmetric paramagnetic phase to the symmetry-breaking (anti-)ferromagnetic phase in the entire excitation spectrum--and not just in the ground state. The topology of the semiclassical energy landscape further determine the quantum corrections which are studied using a spin-wave theory. We obtain analytical predictions that become exact for a chain of long spins.

We derive an analytical description of the ensemble-average dynamics of non-interacting atomic qubits in a spatially uniform, fluctuating magnetic field. The resulting dephasing process is classically correlated and induces an effective atom-atom interaction. This process describes one of the dominant error sources of trapped-ion experiments. We use the derived solution to specify field orientations that preserve any degree of atomic entanglement for all times, and to identify families of states with time-invariant entanglement for arbitrary field orientations. Our formalism applies to arbitrary spectral distributions of the fluctuations. 

Two-dimensional spectroscopy is a powerful tool for the analysis of complex molecular aggregates or solid-state devices. Here we demonstrate how nonlinear spectroscopy can be applied to extract complex dynamical features from quantum many-body systems. The measurable multidimensional spectra contain both dynamical and spectral information that can be used to identify decoherence mechanisms and interactions. In combination with local excitations and readout, information on the spatial distribution of excitations and the shape of eigenstates can be obtained.

Initial correlations between system and environment can significantly influence the open-system dynamics. Certain types of correlations can further lead to a quantum advantage in quantum information protocols. In this talk we present a method that allows to reveal discord-type correlations between system and environment by exploiting their dynamical effect on the reduced dynamics. The method does not require access to the environment, whose nature may be completely unknown to the experimenter. We discuss experimental realizations ranging from two degrees of freedom of single photons and single ions, to chains of 42 ions, as well as theoretical considerations of the method as a local probe in a complex environment.

We develop a statistical approach for the description of the dynamics of complex open quantum systems based on Haar-measure averages over the unitary group. We investigate generic properties of complex open quantum systems, employing arguments from ensemble theory. By averaging the eigenvectors of the Hamiltonian at fixed spectrum, we study the effect of spectral statistics on generic dynamical evolutions with the help of random matrix theory. We present a series of applications of our general results in the context of open quantum systems, including the average quantum dynamical maps and the impact of initial correlations on the dynamics.